Xxxxxx operators on the unit circle and Xxxxxx matrices Sample Clauses

Xxxxxx operators on the unit circle and Xxxxxx matrices. ‌ In this setting, we have an orthogonal projection, P+, sometimes called the Riesz projection, defined as + ∫ P+ : L2(T) → H2 (T) (P+ f)(v) = lim ε→0+ f(z) zdm(z). (3.1.1) T z − (1 − ε)v Let ω ∈ L∞(T) be fixed. We define + + H(ω) : H2 (T) → H2 (T) H(ω)f = P+ΩJP+f, (3.1.2) where (Ωf)(v) = ω(v)f(v), while J is so that Jf(v) = f(v). + Since {vn}n≥0 is an orthonormal basis of H2 (T), then the operator H(ω) can be represented as an “infinite matrix” as follows H(ω)j,k = H(ω)vj, vk L2(T) = Ω(v—j), vk L2(T) = (ω, v )L2(T) = ω^(j + k). j+k ^ This shows that H(ω) is unitarily equivalent to the Xxxxxx matrix Γ (ω) given by Γ (ω) : l2(Z+) → l2(Z+) ^ Σ (Γ (ω))a(k)^= ∞ ω(j + k)a(j), k 0. j=0 ^ ≥